Signal Processing Week 2 Lecture A:¶

Continuous Time Fourier Series (Trig Form)¶

Transforms¶

• Signals are functions that convey information
• Straightfoward ways (like amplitude vs time) of representing a signal may not be the best for showing its properties
• Example: look at frequency over time instead of pressure over time
• Transform: alternate view of a signal
• 6.003: family of transforms known as fourier transforms, where signals are represented as sums of harmonically-related sinusoids

$$f(t) = c_0 + \sum_{k=1}^\infty (c_k \cos(k \omega_0 t) + d_k \sin(k \omega_0 t))$$

• We already dealt with this representation in PSET #1

Key Terms:¶

• A periodic signal repats after some amount of time $T$ where

$$x(t) = x(t+T)$$

• A function periodic in $T$ is periodic in $nT$ for all integer $n$
• The smallest value $T$ for which this holds is called the fundamental period
• The frequency $\omega_0 = \frac{2\pi}{T}$ is the signal's fundamental frequency
• Frequencies are harmonically related if they are integer multiples of some $T$

Sinusoids Note:¶

$$A \cos(\omega t) + B \sin(\omega t) = \big(\sqrt{A^2 + B^2}\big) \cos\Big(\omega t + \tan ^{-1} \big(\frac{-b}{a}\big)\Big)$$

• Can be proven through converting to complex exponential form and taking the real part
• Demonstrates that the sum of sine and cosine terms yields another sinusoid

Integrating over a period¶

$$\int_{t_0}^{t_0+T} \sin\Big(\frac{2\pi k}{T} t\Big) dt = 0$$ $$\int_{t_0}^{t_0+T} \cos\Big(\frac{2\pi k}{T} t\Big) dt = 0$$

• Product Identities

$$\int_{t_0}^{t_0+T} \sin\Big(\frac{2\pi k}{T} t\Big) \cdot \cos\Big(\frac{2\pi m}{T} t\Big) dt=0$$

• True for all integers $k$, $m$

$$\int_{t_0}^{t_0+T} \cos\Big(\frac{2\pi k}{T} t\Big) \cdot \cos\Big(\frac{2\pi m}{T} t\Big) dt$$

• This expression is equal to $T/2$ if $k=m$, otherwise $0$
• Can be proven using trig identities

Fourier Series¶

• Fourier Series are sums of harmonically related sinusoids

$$f(t) = c_0 + \sum_{k=1}^\infty (c_k \cos(k \omega_0 t) + d_k \sin(k \omega_0 t))$$

• $\omega_0$ is the fundamental frequency

• When can we represent a signal as sums of harmonic components:

• Only periodic signals, all harmonics of $\omega_0$ are periodic in $T=2\pi / \omega_0$

The $c_0$ term¶

• The constant $c_0$ is known as the "DC" term
• Constant offset of a signal
• Average vvalue of the signal over one period
• Can be computed by computing the definite integral over a $T$ $f$ is periodic over and dividing by $T$

Computing the other $c_k$ terms¶

$$f(t) = c_1 \cos(\omega_0 t) + d_1 \sin(\omega_0 t) + c_3 \cos(3 \omega_0 t) + d_5 \sin(5 \omega_0 t)$$

• Stratgy for finding $c_3$: multiply by $c_3$ and integrate the result over one period

• Works because all terms in the integral will integrate to 0 by the trig rules from earlier, with the exception of the $c_3 \cos(3 \omega_0 t)$ term

• Rules:

$$c_k = \frac{2}{T} \int_{t_0}^{t_0+T} f(t) \cos\Big (\frac{2\pi k t}{T}\Big)dt$$ $$d_k = \frac{2}{T} \int_{t_0}^{t_0+T} f(t) \sin\Big (\frac{2\pi k t}{T}\Big)dt$$

• Why do we need both $\sin$ and $\cos$ components?
• Allows for representation of both even and odd components of the signal

Practice¶

$$f(t) = 0.8 \sin(6\pi t) - 0.3 \cos(6\pi t) + 0.75 \cos(12 \pi t)$$

• $\omega_0 = 6 \pi$

• $T = \frac{2\pi}{\omega_0}$

• $T = 1/3$

What are the Fourier series coefficients for $f(t)$?

• $c_0 = 0$
• $c_1 = -0.3$
• $c_2 = 0.75$
• $d_1 = 0.8$

Analyzing a Square Wave¶

Which are true:

• $c_k =0$ for all $k$
• True! The function is odd, so it can be analyzed with only $\sin$ terms
• $d_k = 0$ if $k$ is even
• True (can be proven with integration rules)
• $| d_k |$ decreases with $k$
• True
• There are an infinite number of non-zero $d_k$ -True

1 term¶

2 terms¶

10 terms¶

100 terms¶

Why does the corner distort?¶

• With finitely many terms, there are 'overshoots' or 'rings' at discontinuities
• Known as Gibb's Phenomenon
• You can decrease the ringing by descreasing the magnitudes of Fourier coefficients at higher frequencies

Triangular Wave¶

• 6 terms